Beyond the square-root barrier: cubic forms of Perazzo type

1. Problem Statement

The paper addresses the quantitative arithmetic of rational points on a specific singular cubic fourfold $X \subset \mathbb{P}^5$ defined over $\mathbb{Q}$ by the equation:
$$F(x, y) = x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = 0.$$
This variety, known as a Perazzo type cubic, is singular along a plane $\Pi \cong \mathbb{P}^2$ (defined by $y_1=y_2=y_3=0$). The authors aim to establish the Manin conjecture for a crepant resolution $\tilde{X} \to X$ of this variety. Specifically, they seek an asymptotic formula for the counting function $N(B)$ of rational points of height at most $B$:
$$N(B) \sim c B^3 \log B.$$
The central challenge is that standard applications of the circle method to cubic hypersurfaces in $\mathbb{P}^{n-1}$ typically require $n > 6$ to overcome the "square-root barrier" (the limitation where error terms dominate the main term unless the dimension is sufficiently high). For $n=6$ (a fourfold), the circle method usually fails to produce an asymptotic formula with the correct main term unless the variety is smooth or has very specific structures. This paper provides the first unconditional treatment of a reduced and irreducible singular cubic fourfold via the circle method.

2. Methodology

The authors employ the smooth $\delta$-method variant of the circle method, developed by Heath-Brown, which relies on the smooth $\delta$-function technology of Duke, Friedlander, and Iwaniec. The core of their approach involves:

• Decomposition of the Counting Function: The sum is decomposed into a main term $M(B)$ and error terms $E_1(B), E_2(B), E_3(B)$ based on the frequency vectors $(m, n)$ in the Poisson summation formula.
• Analysis of Exponential Sums: A critical component is the analysis of the complete exponential sums:
  $$S_q(m, n) = \sum_{a \pmod q}^* \sum_{x, y \pmod q} e_q(aF(x, y) + m \cdot x + n \cdot y).$$
  The authors prove that for this specific cubic form, these sums exhibit better than square-root cancellation. Specifically, they establish a property denoted as $(\diamond)$: for generic vectors, $|S_p(m, n)| \ll p^3$ (where the trivial bound is $p^{3.5}$ for a cubic in 6 variables). This cancellation is linked to the vanishing of Katz's $A$-number and the specific geometry of the dual variety.
• Handling the Dual Variety: The dual variety $X^*$ is defined by a sextic form $D(m, n)$. The authors show that the main term receives contributions not only from the zero frequency $(m, n) = (0, 0)$ but also from non-zero frequencies lying on the dual variety $D(m, n) = 0$. This split is crucial; the zero frequency contributes $3/4$ of the main term, while the dual variety contributes the remaining $1/4$.
• Advanced Analytic Tools:
  • Hooley's $\Delta$-function: To handle sums over moduli $q$ dividing the dual form $D(m, n)$ without losing a logarithmic factor, the authors utilize bounds for the average of Hooley's $\Delta$-function along polynomial sequences (citing work by la Bretèche–Tenenbaum and Chan–Koymans–Pagano–Sofos).
  • Dirichlet Series and Analytic Continuation: They analyze the Dirichlet series $\xi(s; m, n) = \sum q^{-s} S_q(m, n)$, establishing analytic continuation to the left of the line $\Re(s) = 4$ to extract the main term.
  • Geometry of Numbers: For the contribution from the dual variety ($E_3(B)$), they relate the sum to counting points on lattices $\Lambda(t)$ associated with the singular locus, using successive minima and lattice point counting techniques.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The authors prove that for any smooth weight function $W$ supported away from the singular locus ($y_1y_2y_3 \neq 0$):
$$N(B) = \frac{\sigma_\infty}{\zeta(3)} B^3 \log B + O(B^3 (\log B)^\varepsilon),$$
where $\sigma_\infty$ is the weighted real density associated with $X$.

Verification of Manin's Conjecture

• Resolution: They confirm that this asymptotic formula establishes the Manin conjecture for the crepant resolution $\tilde{X} \to X$.
• Constant Matching: They explicitly calculate the leading constant predicted by Peyre's formula for $\tilde{X}$:
  $$C_{\text{Peyre}} = \alpha(\tilde{X})\beta(\tilde{X})\tau_\infty(\tilde{X})\tau_{\text{fin}}(\tilde{X}).$$
  They show that $\alpha(\tilde{X}) = 1/9$, $\beta(\tilde{X}) = 1$, $\tau_{\text{fin}}(\tilde{X}) = \zeta(3)^{-2}$, and $\tau_\infty(\tilde{X}) = \frac{3}{2}\sigma_\infty$. The product yields exactly $\frac{\sigma_\infty}{6\zeta(3)^2}$, which matches the coefficient derived from the circle method (accounting for the height normalization).

Overcoming the Square-Root Barrier

The paper demonstrates that the "square-root barrier" can be bypassed for specific singular cubic fourfolds. The key mechanism is the super-cancellation in the exponential sums $S_p(m, n)$, which holds for the Perazzo form but not for general cubic forms. This allows the error terms to be controlled sufficiently to extract the $B^3 \log B$ main term.

Classification of Forms with Property $(\diamond)$

The authors identify a class of cubic forms (including the Perazzo form) that satisfy the cancellation property $(\diamond)$. They list examples including:

1. Cubic fourfolds singular along a rational plane.
2. Fourfolds corresponding to pencils of quadric surfaces.
3. Fourfolds defined by trilinear equations.
4. A family $x_1y_1^2 + x_2y_2^2 + x_3y_3^2 = c_1 x_1x_2x_3 + c_2 y_1y_2y_3$.

4. Significance

• Unconditional Proof: Unlike previous works on singular cubic fourfolds (e.g., Wang's conditional work on the diagonal case), this result is unconditional.
• Extension of Circle Method: It significantly extends the applicability of the circle method to singular varieties of lower dimension ($n=6$) where it was previously thought to be inapplicable due to the square-root barrier.
• Dual Variety Contribution: The paper provides a deep insight into how the dual variety contributes to the main term in the circle method. The $3/4 : 1/4$ split of the main term between the zero frequency and the dual variety is a novel phenomenon that highlights the interplay between the geometry of the variety and the analytic properties of the exponential sums.
• Manin Conjecture for Singular Varieties: It provides a rigorous verification of the Manin conjecture for a non-trivial singular cubic fourfold, bridging the gap between toric/singular cases handled by universal torsors and smooth cases handled by the circle method.

In summary, this paper represents a breakthrough in analytic number theory by successfully applying the circle method to a singular cubic fourfold, overcoming traditional dimensional barriers through a sophisticated analysis of exponential sums and the geometry of the dual variety.